Title: Assessing the prognostic ability of the stratified Cox proportional hazards model
1Assessing the prognostic ability of the
stratified Cox proportional hazards model
- Lisa Pennells
- Ian White
- Angela Wood
- Stephen Kaptoge
- Nadeem Sarwar
- The Fibrinogen Studies Collaboration
2Outline
- Background
- The Fibrinogen Studies Collaboration (FSC) The
data - Modelling
- Measures to estimate prognostic ability
- Measures we have chosen to quantify prognostic
value - Adaptation of measures for use with a stratified
model - Adaptation of measures to display their value
over time - Conclusions
3The Fibrinogen Studies Collaboration (FSC)
- Aim to amalgamate available information on the
role of Fibrinogen in Coronary Heart Disease
(CHD) and other Cardiovascular endpoints - Raised usual plasma fibrinogen levels thought to
increase risk of cardiovascular disease. - Combination of data from 31 long-term prospective
studies - 154 211 adults without a history of previous
cardiovascular disease - Median follow up 8.42 years (total 1.38
million person-years) - Several dozen characteristics reported at
baseline including - Established cardiovascular risk factors e.g.
Cholesterol, blood pressure - Main endpoint on which analysis focused First
CHD event (n7118)
Fibrinogen Studies Collaboration, JAMA 2005
294(14)1799-1809.
4Modelling
- Stratified Cox regression
- where
- the probability of surviving beyond time t
given covariates x and strata value k - the baseline survival function for strata
k - In analysis of FSC data we used a Cox model that
was stratified by cohort, sex and in the case of
one study, trial arm
5Hazard ratios obtained from the stratified model
6Assessing the prognostic power of Variables
- The need for measures of Prognostic ability
- Hazard ratios dont tell us how useful a model
is for making predictions - Statistical significance of hazard ratios depends
largely on sample size - Measure needed - Interpretable
- - comparable across data sets of different
sizes - Previous methods used in cardiovascular
epidemiology - Most commonly used Area Under ROC Curve (AUROC)
- Only appropriate for binary data!
- Challenge for FSC
- Identify methods appropriate for use with
survival data - Adapt for use with the stratified model
7Measures Chosen
- Explained variation
- Proportion of variation in the outcome explained
by the covariates in the model. - Range 0 to 1 (values lt 0.4 common for survival
data) - Schemper and Hendersons V
- Designed to be consistent under with random
censoring - Measures of discrimination
- Summarise a models ability to discriminate
between levels of subject risk (or risk ranking) - Harrells C-index
- Equivalent to the AUROC (already well know in CV
epidemiology) - Royston and Sauerbreis D
- Several favourable properties (including
simplicity)
Schemper and Henderson, Biometrics 2000 56(1).
Harrell FE, Jr. et al. JAMA 1982 247(18).
Royston and Sauerbrei. Stat.Med. 2004 23(5).
8V Original
- difference between observed survival and that
predicted using a model without covariates,
averaged a) over all subjects at each failure
point - b) over all failure points (with weights)
- as for but using a model with covariates
- Weights when averaging across failure points to
give and
Schemper M, Henderson R. Biometrics 200056(1)
9V adapted for the stratified model
- Inaccuracies for each individual are determined
according to strata specific predicted survival
probabilities - Inaccuracies are averaged over time using strata
specific weights - V over time
- At each time point of interest (e.g. at 5 year
intervals) only inaccuracies calculated at
failure points occurring before this time are
averaged (using relevant weights)
10Problem with V for the stratified model
- Problem
- V increases initially with time. Hence, if
combining information from studies which differ
in follow up time, inclusion of information from
studies at time points after their study period
has ended leads to downward bias in the value of
V - Solution
- At each time point, only include information
from studies that are still running
11A well behaved V over time
12V from 4 studies with different follow up
13Combining the 4 studies regardless of follow-up
14Effect of combining V from studies of different
follow up
a) Combining regardless of follow-up
b) Using only information from current studies
0.07
15D Original
- Motivation
- To quantify the observed spread of disease risk
across the range of estimated - Calculation
- Fit CPH model
- Transform to give standard normal order
rank statistics (rankits - formed using Bloms
approximation) - Multiply rankits by a factor of to
give zi (i1n subjects) - Fit a CPH model to these values D is the
coefficient of z from this second model - Interpretation
- log hazard ratio comparing two equal-sized
prognostic groups based on dichotomising - a continuous prognostic index ( )
- D adapted for the stratified model
- Replace steps 1 and 4 above by the fitting of a
stratified model
Royston P, Sauerbrei W. Stat.Med. 2004 23(5)
16C-index Original
- Definition The probability that, for a randomly
selected pair of subjects, the person who
fails first has the worse prognosis. - Range 0.5 (discrimination no better than chance
prediction) to 1(perfect discrimination). - Estimation Class all pairs in which the subject
with the shorter participation time fails as - Concordant, agreeing in rank of and order
of failure - Discordant, opposite in rank of and order of
failure - Undecided, tied in either category
- The numbers of class 1, 2 and 3 are then counted
to give , and respectively, and - combined in the calculation of the C-index
Harrell FE, Jr. et al. JAMA 1982 247(18).
17C-index for the Stratified model
- A stratified model is fitted to the data
- Comparison of pairs is restricted to occur within
strata - Numbers of pairs are summed overall
- D and the C-index over time
- A model is fitted to the whole data
- is extracted (and in the case of D
transformed) - The data set is then censored at each time point
before the comparison of pairs (in the case
of C) and the fitting of the second model (in the
case of D)
18Final versions of V, D and the C-index over time
for FSC data
19Conclusions
- For survival data it is important to use a
measure of prognostic value which adequately
handles its characteristics. - Each of V, D and the C-index appeared promising
in principle, for adaptation to the stratified
model. - We found the measure of explained variation V to
be difficult to apply when data are combined from
several studies with differing durations of
participant follow-up. Confidence intervals also
increased throughout follow-up for our data. - The two other measures considered, D and the
C-index, were more applicable in such
circumstances.
20Acknowledgments
- Professor John Danesh (University of Cambridge)
- Members of the Fibrinogen Studies Collaboration
- AMIS JB Kostis, AC Wilson Atherosclerosis Risk
in Communities Study AR Folsom, K Wu, L
Chambless BIP Registry M Benderly, U Goldbourt
Bruneck Study J Willeit, S Kiechl Caerphilly
Study JWG Yarnell, PM Sweetnam (this prospective
cohort study was undertaken by the former UK
Medical Research Council Epidemiology Unit (South
Wales) and was funded by the MRC its data
archive is maintained by the Department of Social
Medicine, University of Bristol) Cardiovascular
Health Study M Cushman BM Psaty, RP Tracy (see
http//chs-nhlbi.org for acknowledgements)
Copenhagen City Heart Study A Tybjærg-Hansen,
ECAT Angina Pectoris Study F Haverkate, MPM de
Maat, SG Thompson Edinburgh Artery Study
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21Composition of V
,